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On Spatial Interpolation of Soil Temperature from Temperature Logs of Monitoring Wells

Posted on: 07.08.2015

Introduction

Non-asymptotic numerical simulation of transient heat transfer requires knowledge of initial conditions. If the heat transfer medium is soil, they typically use temperature logs (depth-temperature tables) of monitoring wells for the initial time point. Then one may either use scattered-data interpolation [1] or solve a steady heat transfer problem with Dirichlet boundary conditions at temperature measurement points. If you choose the second way and the problem is nonlinear, there might be some convergence issues.

Figure 1: Soil Temperature Interpolation in Frost 3D Universal

We use scattered-data interpolation [1] and our method provides a series of useful features, including the following:

In case of single monitoring well, isothermal surfaces follow the terrain, or more precisely, its smoothed (Gaussian blurred [2]) versions. The blur radius is depth-proportional.

In the multi-well case, isothermal surfaces adapt to the terrain. In order to achieve this, we use a straightforward piecewise-linear triangulation-based interpolation [1] along isosurfaces of the smoothed depth field. The smoothed depths of soil points are initially computed as the differences between their smoothed (with Gaussian blur [2] of depth-proportional intensity) elevations and altitudes. Then, if needed, the smoothed depth field is locally corrected to become strictly decreasing with respect to altitude.

Globally, our interpolation is range-restricted. This guarantees that the resulting temperature won't go below the absolute zero. Also, if all measured temperatures are above (below) the freezing point, then the entire interpolation domain will be thawed (frozen).

Along the monitoring wells, the interpolation is shape-preserving, i.e. preserves monotonicity and convexity [3].

Statement of Problem

Let and let be the elevation map determining the part of the surface of the infinitely extended and infinitely deep soil .

We investigate the problem of interpolating the unknown temperature field ( stands for the convex hull of the set of points) from the temperature logs

where are altitudes of the measurement points in the th monitoring well , , . Here is the "reasonable" continuation (see stage 3, below) of the function to the entire plane . The exact definitions of the infinitely extended and deep soil and its surface are the following:

Note that .

Algorithm

One may cope with the given interpolation problem in several stages. Let's list them.

Merge the duplicate wells and remove the empty (containing no measurement points) ones. If there are no wells now, stop. Renumber the remaining wells, redefine .

Merge the duplicate measurement points (for multiple temperature values, use their arithmetic mean as the new temperature). Renumber the updated measurement points in and redefine , .

Construct the convex hull and the Delaunay triangulation of the same set of points .

In each fixed point (let's denote its smoothed depth with ), compute the temperature using the following algorithm. Let be the closest to point of the convex hull . In particular, if , then . If the point lies in some non-degenerate Delaunay triangle , and therefore there exist such uniquely determined scalars (barycentric coordinates) that and , then put

There's also the case when the Delaunay triangulation is empty (contains no non-degenerate triangles). In this situation (keep in mind that ), the point lies on some non-degenerate segment (), i.e. , , where the choice of the scalars is unique. Put

It is clear from construction that isosurfaces of the resulting temperature field adapt to the terrain, but, as the depth increases, this adaptation weakens monotonically.